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\title{Analysis}
\author{Last Updated: September 2002}
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\paragraph{Real Analysis:}
The real number system; Metric spaces -- topology, completeness,
connectedness, compactness; sequences and series, Cauchy sequences;
continuity, uniform continuity; pointwise and uniform convergence of
functions; definition and properties of the Riemann
integral; uniform convergence and approximation, the
Stone--Weierstrass Theorem; Ascoli's Theorem.
\paragraph{Complex Analysis:}
Analytic and harmonic functions; Cauchy-Riemann equations; power
series; Cauchy's Theorem,
Cauchy's Formula; Liouville's Theorem; Taylor's Theorem; The Maximum
Modulus Theorem; Morera's Theorem; the theory of isolated
singularities: Cassardi-Weierstrass Theorem; Laurent expansion, the
residue theorem, application to definite integrals; elementary
functions and their mapping properties; analytic continuation; Riemann mapping theorem.
\paragraph{Measure Theory:}
Lebesgue measure; general measure and integration; Carath\'eodory's
Theorem; convergence in
mean; convergence in measure; the Monotone Convergence Theorem;
Fatou's Lemma; the Dominated Convergence Theorem; H\"older's inequality;
$L^p$-spaces; Fubini's Theorem.
\paragraph{Functional Analysis:}
Elementary Banach space and Hilbert space theory to include: linear
functionals; the Hahn-Banach Theorem; dual spaces; the Uniform
Boundedness Principle; the Open Mapping Theorem; the Closed Graph
Theorem; the Riesz-Fischer Theorem; the Riesz Representation Theorem;
orthonormal bases; bounded linear transformations; compact operators
and the spectral theorem for compact self-adjoint operators.
\paragraph{NOTE 1.}
You should know precise statements of definitions and theorems
together with relevant examples and counterexamples.
\paragraph{NOTE 2.}
Differentiable manifolds are important in analysis but are adequately
covered in the topology certification examination.
\subsubsection*{REFERENCES}
\paragraph{Real Analysis:} Both Russell Gordon's \emph{Real Analysis:
a first course} and chapters 1--8 of Rudin's \emph{Principles of
Mathematical Analysis} are pretty complete. A more general,
topological emphasis is given in Chapters 1--8 of Manfred Stoll's
\emph{Introduction to Real Analysis.}
\paragraph{Complex Analysis:} Chapters I--IV and VII \S1--4 of
Conway's (John B.) book \emph{Complex
Analysis~I} is a pretty complete treatment. Brown \& Churchill's
\emph{Complex Variables and Applications} is a standard treatment,
but lacks sophistication. Chapter~10 of Rudin's \emph{Real \&
Complex Analysis} is a very terse and sophisticated supplemental
source.
\paragraph{Measure Theory:} Chapters 1--3, 6 and 7 of
Folland's \emph{Real Analysis} are very good. Another good source are
chapters 1--4, 6, 11 and 12 of Royden's \emph{Real Analysis}.
(Chapters~1--4 on Lebesgue measure should be read by everybody.) A
more sophisticated treatment can be found in chapters 1--3 and 8 of
Rudin's \emph{Real \& Complex Analysis}. See also Chapter 10 of Stoll's book on real
analysis above.
\paragraph{Functional Analysis:}
Chapter 5 of Folland's \emph{Real Analysis} and/or chapters 1--3 of
Conway's \emph{A Course in Functional Analysis} are a good place to
start --- as are chapters 9 and 10 of Royden's \emph{Real Analysis} and
chapters 0,1 and 3 of Robert J. Zimmer's \emph{Elements of Functional
Analysis}.
For more adventure, chapters 4 and 5 of Rudin's \emph{Real \& Complex
Analysis} and chapters 1--3 of Pedersen's \emph{Analysis Now} are
sophisticated, but excellent resources.
\paragraph{Comments:} No one is expected to read and absorb all of the
references listed above. We've provided these books as suggested
resources, and you may prefer to use different texts. You are
encouraged
to discuss your reading lists with potential members of your
certification committee.
% Rudin's \textit{Principles of Mathematical Analysis} covers the real
% analysis topics in an elementary way. For a more sophisticated
% treatment, see Hewitt and Stromberg's \textit{Real and Abstract
% Analysis} which also contains lots of measure theory.
% Knopp's \textit{Theory of Functions, Volume 1} is an easy
% introduction---good for independent study---to complex analysis. Of
% the many other texts Marsden's \textit{Basic Complex Analysis} is
% quite adequate. Ahlfors' \textit{Complex Analysis} is also good.
% Halmos' \textit{Measure Theory} is the standard treatise, but the
% syllabus is covered also in Hewitt and Stromberg, in Rudin's
% \textit{Real and Complex Analysis} and, in less detail, in Goffman and
% Pedrick's \textit{First Course in Functional Analysis}.
% Rudin's \textit{Real and Complex Analysis} and \textit{Functional
% Analysis} cover the functional analysis syllabus. Yosida's
% \textit{Functional Analysis} and Kato's \textit{Perturbation Theory}
% are also useful.
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